A construction of a $\lambda$-Poisson generic sequence

Abstract:

Years ago Zeev Rudnick defined the $\lambda$-Poisson generic sequences as the infinite sequences of symbols in a finite alphabet where the number of occurrences of long words in the initial segments follow the Poisson distribution with parameter $\lambda$. Although almost all sequences, with respect to the uniform measure, are Poisson generic, no explicit instance has yet been given. In this note we give a construction of an explicit $\lambda$-Poisson generic sequence over any alphabet and any positive $\lambda$, except for the case of the two-symbol alphabet, in which it is required that $\lambda$ be less than or equal to the natural logarithm of $2$. Since $\lambda$-Poisson genericity implies Borel normality, the constructed sequences are Borel normal. The same construction provides explicit instances of Borel normal sequences that are not $\lambda$-Poisson generic.